Horner's Method - Description of The Algorithm

Description of The Algorithm

Given the polynomial

where are real numbers, we wish to evaluate the polynomial at a specific value of, say .

To accomplish this, we define a new sequence of constants as follows:

\begin{align}
b_n & := a_n \\
b_{n-1} & := a_{n-1} + b_n x_0 \\
& {}\ \ \vdots \\
b_0 & := a_0 + b_1 x_0.
\end{align}

Then is the value of .

To see why this works, note that the polynomial can be written in the form

Thus, by iteratively substituting the into the expression,


\begin{align}
p(x_0) & = a_0 + x_0(a_1 + x_0(a_2 + \cdots + x_0(a_{n-1} + b_n x_0)\cdots)) \\
& = a_0 + x_0(a_1 + x_0(a_2 + \cdots + x_0(b_{n-1})\cdots)) \\
& {} \ \ \vdots \\
& = a_0 + x_0(b_1) \\
& = b_0.
\end{align}

Read more about this topic:  Horner's Method

Famous quotes containing the words description of the, description of and/or description:

    Why does philosophy use concepts and why does faith use symbols if both try to express the same ultimate? The answer, of course, is that the relation to the ultimate is not the same in each case. The philosophical relation is in principle a detached description of the basic structure in which the ultimate manifests itself. The relation of faith is in principle an involved expression of concern about the meaning of the ultimate for the faithful.
    Paul Tillich (1886–1965)

    Why does philosophy use concepts and why does faith use symbols if both try to express the same ultimate? The answer, of course, is that the relation to the ultimate is not the same in each case. The philosophical relation is in principle a detached description of the basic structure in which the ultimate manifests itself. The relation of faith is in principle an involved expression of concern about the meaning of the ultimate for the faithful.
    Paul Tillich (1886–1965)

    The Sage of Toronto ... spent several decades marveling at the numerous freedoms created by a “global village” instantly and effortlessly accessible to all. Villages, unlike towns, have always been ruled by conformism, isolation, petty surveillance, boredom and repetitive malicious gossip about the same families. Which is a precise enough description of the global spectacle’s present vulgarity.
    Guy Debord (b. 1931)